High order congruences for M-ary partitions

For a sequence \(M=(m_{i})_{i=0}^{\infty }\) $M={(m_i)}_{i=0}^{\infty }$ of integers such that \(m_{0}=1\) $m_0=1$ , \(m_{i}\ge 2\) $m_i\ge 2$ for \(i\ge 1\) $i\ge 1$ , let \(p_{M}(n)\) $p_M(n)$ denote the number of partitions of n into parts of the form \(m_{0}m_{1}\cdots m_{r}\) $m_0{m}_1\cdots m_r$ . In this paper we show that for every positive integer n the following congruence is true: \(\begin{aligned} p_{M}(m_{1}m_{2}\cdots m_{r}n-1)\equiv 0\ \ \left( \textrm{mod}\ \prod _{t=2}^{r}\mathcal {M}(m_{t},t-1)\right) , \end{aligned}\) $$\begin{array}{c}\hfill p_M(m_1{m}_2\cdots m_{r}n-1)\equiv 0\left(\text{mod},,\prod _{t=2}^r,\mathcal{M},(m_t,t-1)\right),\end{array}$$ where \(\mathcal {M}(m,r):=\frac{m}{\textrm{gcd}\big (m,\textrm{lcm}(1,\ldots ,r)\big )}\) $\mathcal{M}(m,r):=\frac{m}{\text{gcd}(m,\text{lcm}(1,\dots ,r))}$ . Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for m-ary partitions found by Andrews, Gupta, and Rødseth and Sellers.